3D Geometry
Perpendicular from Point to Line PQ — l²
nta_pyq_2024_jan
Grade 12
Question:
A line with direction ratios $2,1,2$ meets the lines $x=y+2=z$ and $x+2=2y=2z$ respectively at the points $P$ and $Q$. If the length of the perpendicular from the point $(1,2,12)$ to the line $PQ$ is $l$, then $l^2$ is
Step-by-Step Solution
Key Concept: Line 1: $x=y+2=z\Rightarrow(t,t-2,t)$. Line 2: $x+2=2y=2z\Rightarrow(2s-2,s,s)$. Direction $PQ\parallel(2,1,2)$: $PQ=(2s-2-t,s-t+2,s-t)\propto(2,1,2)$. Solve for $s,t$.
$l^2=65$.
Correct Answer: 65